Source-receptor matrix calculation with a Lagrangian particle dispersion model in backward mode

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particle dispersion model in backward mode

P. Seibert, A. Frank

To cite this version:

P. Seibert, A. Frank. Source-receptor matrix calculation with a Lagrangian particle dispersion model

in backward mode. Atmospheric Chemistry and Physics, European Geosciences Union, 2004, 4 (1),

pp.51-63. �hal-00295385�

Atmos. Chem. Phys., 4, 51–63, 2004 www.atmos-chem-phys.org/acp/4/51/ SRef-ID: 1680-7324/acp/2004-4-51

Atmospheric

Chemistry

and Physics

Source-receptor matrix calculation with a Lagrangian particle

dispersion model in backward mode

P. Seibert and A. Frank

Institute of Meteorology and Physics, University of Natural Resources (BOKU), Wien, Austria Received: 17 June 2003 – Published in Atmos. Chem. Phys. Discuss.: 29 August 2003 Revised: 12 December 2003 – Accepted: 10 January 2004 – Published: 23 January 2004

Abstract. The possibility to calculate linear-source recep-tor relationships for the transport of atmospheric trace sub-stances with a Lagrangian particle dispersion model (LPDM) running in backward mode is shown and presented with many tests and examples. This mode requires only minor modifications of the forward LPDM. The derivation includes the action of sources and of any first-order processes (trans-formation with prescribed rates, dry and wet deposition, ra-dioactive decay, etc.). The backward mode is computation-ally advantageous if the number of receptors is less than the number of sources considered. The combination of an LPDM with the backward (adjoint) methodology is espe-cially attractive for the application to point measurements, which can be handled without artificial numerical diffusion. Practical hints are provided for source-receptor calculations with different settings, both in forward and backward mode. The equivalence of forward and backward calculations is shown in simple tests for release and sampling of particles, pure wet deposition, pure convective redistribution and real-istic transport over a short distance. Furthermore, an applica-tion example explaining measurements of Cs-137 in Stock-holm as transport from areas contaminated heavily in the Chernobyl disaster is included.

1 Introduction

The source-receptor (s–r henceforth) relationship is an im-portant concept in air quality modelling. It describes the sen-sitivity of a “receptor” element y to a “source” x. The re-ceptor could be, for example, the average concentration of a certain atmospheric trace substance in a given grid cell dur-ing a given time interval. It could also be a deposition value, Correspondence to: P. Seibert

(www.boku.ac.at/imp/envmet/seibert mail.html)

and instead of a grid volume average it could be the value at a measurement station. The source could be a point, area or volume source acting during a specified time interval in a specific location.

The s–r relationship can be linear or nonlinear. In the non-linear case, it is defined by the partial derivative ∂y/∂x at a given state of the system. This paper deals with linear s–r relationships, because standard Lagrangian particle disper-sion models cannot simulate nonlinear chemical reactions. However, all the other processes occurring during the atmo-spheric transport of trace substances are linear: advection, diffusion, convective mixing, dry and wet deposition, and ra-dioactive decay. First-order chemical reactions, where the reaction rates can be prescribed, are also linear. In the linear case, the s–r relationship reduces to the simple expression y/x. Thus, linear s–r relationships can be calculated easily with any dispersion model M : x→y.

In typical applications, there are many receptor elements and/or many sources. We will denote them by vectors y and x, respectively. The s–r relationships can be written as a s–r matrix (also abbreviated as SRM) M whose elements milare

defined by

mil= yl xi .

In the case of gridded data, the temporal and all the three spatial dimensions are combined into each of the indices (i=1, .., I for the sources, and l=1, .., L for the receptors). This notation is in agreement with inverse modelling stud-ies. In the remaining sections of this paper, receptors are specified as mixing ratios or concentrations and as such de-noted with symbols χ and c. Sources are specified as mass flux densities or masses and denoted by ˙qor Q, respectively. Thus, the s–r relationship can be denoted by ∂χ /∂ ˙qor simi-lar expressions.

The advantage of knowing M is that for a given source vector, the resulting receptor values can be obtained by a simple matrix-vector multiplication, avoiding the evaluation of the whole numerical model of the transport and dispersion processes. This is useful for scenario considerations. Fur-thermore, the SRM contains detailed budget information; it describes which receptors are affected by a specific source element (by the corresponding column of M) and which sources are contributing to a specific receptor (by the row of M) corresponding to the receptor). The SRM gives a quite comprehensive description of a transport and disper-sion problem. Its determination is therefore also an important intermediate step for solving optimisation problems, such as the minimisation of environmental impact under defined abatement costs, or inverse modelling to determine sources from given measurements.

The determination of the SRM is much more costly in terms of computing than a single simulation with a given source configuration because it has to be either rerun for each source element, or (more efficiently but also requir-ing more computer memory) a separate species has to be tracked for each source element (in forward mode). In the case of I >L (more source elements than receptor elements), receptor-oriented methods are more efficient than forward methods, and vice versa. These receptor-oriented methods are known as adjoint or backward methods. They need a number of simulations or species equal to the number of receptors (whereas forward methods need as many simula-tions or species to be tracked as there are sources). For the case of Eulerian models, many authors have pointed this out and applied adjoint Eulerian models for such tasks (e.g. Uliasz (1983), Uliasz and Pielke (1992), Robertson and Pers-son (1992), Giering (1999), Hourdin and Issartel (2000)). For Lagrangian particle models, Thomson (1987) and Flesch et al. (1995) have shown that the Lagrangian particle model is basically self-adjoint (except the sign of advection). Flesch et al. (1995) introduced backward integration of such a model as a tool to determine so-called footprints for measurements of micro-meteorological fluxes. Footprint modelling includ-ing the backward methodology has become an established tool in the micrometeorological community for interpretation of both fluxes and scalar quantities and the planning of mea-surements (Schmid, 2002; Kljun et al., 2002).

Seibert (2001) introduced this concept in air pollution modelling and for arbitrary volume sources. However, all these papers were limited to conservative tracers, without considering sinks or any other first-order processes.

The present papers shows that a Lagrangian particle dis-persion model can be used in a backward-running, receptor-oriented mode to determine s–r relationships including any such first-order processes. The concept is valid for source and receptor volumes of arbitrary shape and location. We apply it to problems ranging from mesoscale to global. The

derivation is followed by sample calculations illustrating practically the equivalence between forward and backward-mode calculations and an application example. Especially Test 3 and the associated Fig. 1 may serve to help understand the principle of s–r relationships and their calculation by for-ward or backfor-ward calculations.

Methods presented here have been integrated into the model Flexpart1 (Stohl et al., 1998), which was also used for the sample calculations. This model is presently used by about 15 groups all over the world and it is freely available. We feel therefore that it is not inappropriate to include some remarks on a more technical level which are related specifi-cally to this model.

To avoid misunderstandings, we would like to point out that the Lagrangian particle dispersion model should not be confused with other types of Lagrangian models such as stan-dard trajectory models or box models tied to such stanstan-dard trajectories. The LPDM is a fully three-dimensional model including the mean horizontal and vertical wind and three-dimensional turbulence.

2 Theory

2.1 Forward simulations with an LPDM

In the forward mode, particles are released in prescribed source regions, and transported with the mean and a stochas-tic turbulent velocity field. They carry a mass depending on source strength and particle release rate which can be altered by processes such as dry and wet deposition or decay. Grid-ded concentrations are calculated by summing up the masses of all the particles in a grid cell, and dividing the sum by the cell’s volume. More refined formulations, used for exam-ple in the model Flexpart (Stohl et al., 1998) may apply ker-nels (Haan, 1999) to determine gridded and especially single-receptor point concentrations.

In the backward mode, used for receptor-oriented mod-elling, the same formalism and computer model are applied, but the particle trajectories are integrated backward in time, using a negative time step. However, these particles are only a means to determine the trajectories and to probe the pro-cesses experienced by substances transported. Furthermore, the approach is based on mass mixing ratios rather than mass concentrations, as this is a conservative quantity.

1_{more information at http://www.forst.tu-muenchen.de/EXT/}

LST/METEO/stohl/flexpart.html

P. Seibert and A. Frank: Source-receptor matrix calculation with a Lagrangian particle dispersion model 53 2.2 Lagrangian, receptor-oriented view of transport and

dispersion

In the following, a mathematical description of the transport and dispersion process in the atmosphere including sources and linear processes (such as, e.g. deposition or decay) is given in a Lagrangian framework with a receptor-oriented view. The result is compared to the formalism employed in typical Lagrangian particle dispersion models.

We consider the individual change of the mass mixing ra-tio χ of an atmospheric trace substance. As we are in a La-grangian frame of reference, there are no advective changes and χ is affected only by sources ˙q or by any linear pro-cesses which are proportional to χ . Nonlinear propro-cesses are not considered in this paper. The sources ˙qare given as mass per volume and time. Thus,

dχ (t )

dt =

1

ρ(t )q(t ) + α(t )χ (t ),˙ (1)

where α is the net decay or build-up rate “constant” and t is time.

This ordinary differential equation can be solved, yielding the mixing ratio at time t and location r∗:

χ (r∗, t ) = χ0 exp  − Z t 0 α[r(t0), t0]dt0  + Z t 0 ˙ q[r(t0), t0] ρ[r(t0_{), t}0_] exp  − Z t t0 α[r(t00), t00]dt00  dt0, (2)

where {r(t0_{), t}0 _{≤ t }is the back trajectory arriving at point} r∗ _{at time t, and χ}

0is the initial mixing ratio (background)

at t =0.

Let us write as an abbreviation

p(t0) =exp  − Z t t0 α[r(t00), t00]dt00  (3)

and call it the transmission function because it determines the fraction of material which is transmitted along a trajectory to the receptor. Let us further write in abbreviated form ˙q(t0) and ρ(t0_{)with the meaning that it denotes ˙qand ρ along the}

trajectory as explained above. Equation (2) then reads

χ (r∗, t ) = χ0p(0) + Z t 0 ˙ q(t0) ρ(t0₎p(t 0_{) dt}0_{. (4)}

Equations (2) and (4) are valid for instantaneous mixing ratios, affected by turbulent fluctuations. The mean mixing ratios should be obtained as ensemble averages. In practi-cal applications, measured concentrations at monitoring sta-tions represent nearly a point in space but an average in time.

P. Seibert and A. Frank: Source-receptor matrix calculation with a Lagrangian particle dispersion model

7

Table 2. Results of release and sampling test.

type of calculation s–r relationship [s]

analytical 43,200

forward 43,233

backward 43,233

Number of particles: 1000

Synchronisation time interval: 300 s Table 3. Results of wet scavenging test.

type of calculation s–r relationship [s]

analytical 2668

forward 2666

backward 2664

Number of particles: 1000

Synchronisation time interval: 300 s

Scavenging parameters:ª¬«l­„®°¯²± šF³W´‚µ=«l±g¶· mmš› h Precipitation rate [mm hš¸› ]: 1.9 + 0.1 INT (time / 300 s)

3.2 Test 2: Wet scavenging

This test is the same as Test 1, except that a strong wet

scavenging is switched on with a prescribed, linearly

vary-ing precipitation rate. In the available versions of Flexpart,

wet scavenging acts on all particles regardless of their height.

Thus, the true s–r relationship could be calculated

analyti-cally. For a description of the wet scavenging algorithm in

Flexpart, see Stohl and Seibert (2002). Furthermore, tests

were conducted for stationary precipitation fields with

con-vective precipitation only, large-scale precipitation only, or

mixed precipitation, all with a total cloud cover of 1. In all

these tests, differences were at most a few seconds (Table 3).

3.3 Test 3: Realistic scenario for short-distance transport

In this test, particles were released in the same grid volume

as in the previous tests, however, they were sampled in

an-other grid volume two degrees to the south-east, during the

same 24 h as the release. The co-ordinates of the grid centres

were 57

N, 20

E and 59

N, 18

E. Real meteorological fields

(ECMWF analysis, 1

horizontal resolution, 60 layers, every

3 h) were used as input, and three different types of tracers

were simulated: conservative, short-lived (‘radioactive noble

gas’, half-life 0.5 d), and a tracer subject to wet scavenging

(‘aerosol’, scavenging properties as in Test 2). This test was

carried out for the days 1–20 October 2000. Convection was

switched off (note that this refers only to vertical

redistri-bution through convection, convective precipitation is taken

from ECMWF analysis and is always considered). In this

case, fluctuations of the concentrations occur as a result of

the stochastic nature of the LPDM.

Figure 1 illustrates the test as well as the forward and

back-Fig. 1. Example of the forward, source-oriented mode (top) and the backward, receptor-oriented mode (bottom) calculations performed for Test 3: realistic short-range transport of a conservative tracer. Calculations were performed for 11 October 2000 over the Baltic Sea and its surroundings (coast lines and borders [as of 1986] are shown by thin lines). The light square indicates the release areas in each mode. Sampling is performed on the whole grid, but only the sample of the bold square can be used for comparison of the forward-mode and backward-mode calculation of the s–r relation-ship. The length of each square is 1¹ . The parameter shown is the s–r relationship [s], calculated with the mode represented by green squares in Figure 2.

ward calculations modes for the s–r relationship in general.

In the forward mode, particles are released from the area of

the light square, which is the source in the s–r relationship

considered in this test, and tracked as they are transported by

the atmospheric flow. The difference of this mode to

regu-lar dispersion modelling is only that results have been scaled

according to Eq. 18. In the backward mode, particles are

re-leased from the area of the receptor in the s–r relationship

(which is marked by the bold square in the forward-mode

www.atmos-chem-phys.org/0000/0001/

Atmos. Chem. Phys., 0000, 0001–12, 2003

Fig. 1. Example of the forward, source-oriented mode (top) and the

backward, receptor-oriented mode (bottom) calculations performed for Test 3: realistic short-range transport of a conservative tracer. Calculations were performed for 11 October 2000 over the Baltic Sea and its surroundings (coast lines and borders (as of 1986) are shown by thin lines). The light square indicates the release areas in each mode. Sampling is performed on the whole grid, but only the sample of the bold square can be used for comparison of the forward-mode and backward-mode calculation of the s–r relation-ship. The length of each square is 1◦. The parameter shown is the s–r relationship [s], calculated with the mode represented by green squares in Fig. 2.

Thus, we need to average χ in time. If the averaging time exceeds well the time scale of the turbulent fluctuations of χ (and it does so in trace substance monitoring), the tem-poral average of the instantaneous values is the same as the

temporal average of the respective ensemble means ( ¯χ =hχ i, where the overbar indicates temporal averaging and h i the ensemble averaging). Similar considerations hold for aircraft measurements which are typically line integrals or satellite measurements representing volume integrals. For the mean (time-integrated) mixing ratio ¯χwe obtain

¯ χ (t1, t2) = 1 t2−t1 Z t2 t1 χ (r∗, t ) dt = 1 t2−t1 Z t2 t1  χ0(t ) p(t,0) + Z t 0 ˙ q(t, t0)p(t, t0) ρ(t, t0₎ dt 0  dt, (5)

where a second time variable t appears in ˙q, ρ, and p to denote trajectories arriving at different times t .

We now introduce a discretisation. Arrival time t is dis-cretised into J time slots each represented by one back tra-jectory. The trajectories arrive at equal intervals between t1

and t2and are designated by index j . Space is gridded with

index i. Furthermore, we need a discretisation of t0, denoted by the index n. This leads to

¯ χ ≈ χ0p(0) + 1 J X j X i X n  ˙qin ρin pj n1tij n0  (6)

with 1t_{ij n}0 being the residence time of trajectory j in the spatio-temporal grid cell (i, n). Note that the integration along the trajectory (a line integral in the joint time-space di-mension) has been replaced by a double sum, one over time and one over space, with the residence time being not only the discrete representation of dt but also the indicator of the trajectory movement. As the source function ˙q/ρ depends only on the time when the back trajectory passes but not on the time when it arrives at the receptor, Eq. (6) can be rear-ranged to (writing = instead of ≈ for simplicity)

¯ χ = χ0p(0) + X i X n " ˙ qin ρin 1 J X j (pj n1tij n0 ) # . (7)

We are now able to calculate the s–r relationship: ∂ ¯χ ∂ ˙qin = 1 J X j pj n1t_{ij n}0 ρin . (8)

In the case of α=0 and thus p=1, the s–r relationship for a source expressed as mass mixing ratio ( ˙q/ρ) is the aver-age residence time of the back trajectories in the grid cell of the source under consideration, in agreement with the earlier derivation for this specific case (Seibert, 2001). In the gen-eral case, the residence times are to be “corrected” with the transmission function p.

2.3 Computation of transmission-corrected residence times with a Lagrangian particle dispersion model

The transmission-corrected residence times, yielding the s–r relationships according to Eq. (8), can be calculated easily with a Lagrangian particle dispersion model running back-ward in time. As discussed in in Sect. 2.1, regular LPDMs generate particles at source locations, follow their forward trajectories, and let different processes such as deposition act on the mass attached to the computational particle. By re-versing the sign of the advection, the trajectories can easily be calculated backward. The LPDM Flexpart (Stohl et al., 1998) has this capability already built in. It required a few other modifications in the code, allowing input and output to occur in reverse temporal direction, etc. The first-order processes that have been summed up in the function α re-main the same as in a forward calculation, in the sense that the mass will be, e.g. reduced by decay whether going back-ward or forback-ward. Due to the linear nature of the term αχ , the direction of the time-stepping does not matter. If the back-ward calculation reverses the sign of the time step instead of the advection velocities (as does Flexpart), the integration of processes related to α must be carried out with the absolute value of the time step.

Turbulence is represented in LPDMs by a stochastic pro-cess with a prescribed pdf for the turbulent wind compo-nents (or their random parts, if a model with autocorrelation is used). Let P (vi)be the probability density function for

the velocity component vi. In order to calculate back

trajec-tories, this probability must be replaced by P (−vi)(Flesch

et al., 1995). If only a symmetric pdf is implemented, for ex-ample a Gaussian distribution as in Flexpart, no code modi-fication is required. An integration with a negative time step will give the correct backward trajectories. However, the cloud convection scheme in Flexpart has different probabil-ities for upward and downward movements and thus needed to be adapted (see Sect. 2.6).

The statistics of the turbulence do not depend on the di-rection of a trajectory. In mathematical terms, this means that the Lagrangian equation of transport in a turbulent fluid, in the absence of asymmetric random motions as discussed above, is self-adjoint with the exception of a change of sign in the mean-flow advection (Flesch et al., 1995).

Let us now see how the standard output of a model like Flexpart can be used to extract these transmission-corrected residence times. Flexpart writes out mass concentrations ¯c, averaged over a grid element (index i) and a time interval 1T (index n). They are computed as

¯ cin= 1 ViN∗ n+N∗ X n∗_=n+1 J X j =1 µj n∗f_{ij n}∗, (9)

P. Seibert and A. Frank: Source-receptor matrix calculation with a Lagrangian particle dispersion model 55 where N∗ _{is the number of samples taken within 1T , and}

the time-dependent particle mass µj n∗is

µj n∗ =µ₀p_{j n}∗ (10)

with µ0 being the initial mass associated with a particle

(which is a constant for each release in Flexpart). fij n∗ is

a function that determines if and how much a particle con-tributes to a given grid cell. In the simplest case, fij n∗would

be 1 if trajectory j is in grid cell i at time step n∗and 0 other-wise; in Flexpart, it is used with a kernel formulation which distributes the mass over four grid elements as a measure to counteract the numerical errors due to the limited number of trajectories. Combining Eqs. (9) and (10), we obtain for the LPDM output: ¯ cin = µ0 ViN∗ n+N∗ X n∗_=n+1 J X j =1 pj n∗f_{ij n}∗. (11)

The calculation of the s–r relationship with the LPDM requires, of course, that the temporal discretisation of the source as introduced in the previous section matches the time intervals used for averaging the output in the LPDM. For a numerical representation, we can adopt the same strategy as in Flexpart and divide the time interval of interest into N∗ small time slices, using the same function f for each of these intervals. Thus, Eq. (8) becomes

∂ ¯χ ∂ ˙qin = 1T N∗ n+N∗ X n∗_=n+1 1 J J X j =1 pj n∗f_{ij n}∗ ρin∗ . (12)

Considering that J is independent of n∗, we may write ∂ ¯χ ∂ ˙qin = Vi1T J µ0  cin ρin  =Vi1T µt ot  cin ρin  (13)

with µt ot as abbreviation for the total initial mass released in

the time interval between t1and t2. (cin/ρin)differs from the

standard output of the LPDM as given in Eq. (11), ¯cin, only

in the density by which the contributions of the single parti-cles in an output grid cell have to be divided. This division is necessary because we have expressed the source in mass units. If, instead, the source would be specified as alteration rate of the mixing ratio, the standard output ¯cincould be used

directly. Note that in this case the dimension of the r.h.s. is time. In any case, it is obvious that the standard LPDM pro-vides a way to obtain s–r relationships by running it back-ward in time with very few modifications.

A key assumption made to arrive at this result is that par-ticles carry mixing ratios rather than masses. This is equiva-lent to the change from ∇ · ρχ v to v · ∇ρχ between forward and adjoint Eulerian transport models (Elbern and Schmidt,

Table 1. Implementation scheme for different concentration units

in an LPDM. “mass” refers to mass units (kg m-3_s-1_{for the source,}

kg m-3for the receptor), “mix” to mixing ratio units (s-1for the source, dimensionless for the receptor). Given are in the column “release” the factor with which the initial mass of a particle has to be multiplied, and in the column “sampling” the factor to multiply particles’ masses with at the moment of sampling. Note that in backward simulations the release takes place at the receptor and the sampling at the source. The respective units of the s–r relationship are given in the last column. The first value (C:) is for sources given as rates per time (and volume in the case of mass), the second one (I:) for instantaneous local sources (for example, a mass in units of kg). They are the same for forward and backward calculations!

1999). At both ‘ends’ of a simulation, at the source and at the receptor, mass units may be desired. This is true for for-ward as well as for backfor-ward simulations. The conversion between both forms is accomplished by multiplication or di-vision with the local air density. As density can vary within a source or receptor volume, this conversion is best done particle-wise. Work on this paper and associated tests as well as ongoing practical applications have shown that keeping maximum flexibility in an LPDM code with respect to mix-ing ratio versus mass units is desirable. Table 1 indicates how to set up a model for all the possible combinations. However, additional scalings are needed if the s–r relationship is to be derived from forward calculations and defined in the same way as done here (see Eq. 18).

2.4 Practical considerations

For gases, volume mixing ratios are a more popular unit than mass mixing ratios. The conversion is simple: multiplica-tion with the ratio of the molecular mass of air to that of the gaseous trace substance or vice versa.

Another widespread practice is to report mass concen-trations at standard temperature and pressure (STP). In this case, Table 1 can be applied with local air density ρ replaced by the air density at STP. As this is a constant, it is possible, for example, to do a backward simulation for receptor units of mixing ratio, and to divide the resulting s–r relationship by ρ

STPlater in order to use it in conjunction with measurements

given in STP concentrations.

Another issue is that often we would be interested in an area source at ground level. To accomplish this, let us con-sider any area A and a vertical layer adjacent to the ground of thickness d as the source volume. Diluted over this layer, an area source ˙qA(in kg m-2s-1) would correspond to a volume

source (in kg m-3_s-1_{) ˙q= ˙q} Ad−1. Thus, ∂ ¯χ ∂ ˙qA = 1 d ∂ ¯χ ∂ ˙q. (14)

Obviously, as ∂ ¯χ /∂ ˙qAcontains the (density-weighted)

resi-dence time in the layer, it will, for well-mixed conditions and a layer not too deep, be proportional to d, so that ∂ ¯χ /∂ ˙qAis

independent of d. The condition of being well-mixed is usu-ally fulfilled within the boundary layer after some travel dis-tance, except for a strongly depositing substance under stable conditions. To obtain good numerical stability, one should not choose the depth of the lowest layer too small, and ex-cept for the above-mentioned circumstances it can be cho-sen to be the typical minimum mixing height expected in the model. It is also interesting to note that the residence time in this layer of a single trajectory being reflected on the ground with vertical velocity w is 2d/w. Thus, the s–r relationship for a conservative tracer (p=1) and an area source is

∂ρχ ∂ ˙qA = 1 J d J X j =1 2d fj wj =2J ∗ J w −1_{, (15)}

twice the fraction of trajectories impinging on the ground weighted with the inverse of their vertical velocity, a con-clusion already drawn by Flesch et al. (1995).

To complete the practical formulae, the s–r relationship for a source in kg s-1_{(point source, one-grid cell source), having}

units of s kg-1_{, would be} ∂ ¯χ Vi∂ ˙qin = 1T µt ot  cin ρin  , (16)

and for an instantaneous point source (or the total source for a given time interval) in kg, having units of kg-1_,

∂ ¯χ 1T Vi∂Qin = 1 µt ot  cin ρin  . (17)

Instead of expressing ambient concentrations and sources in kilogramme, one can always substitute that by Becquerel if radioactivity is under consideration.

Flexpart and other LPDMs allow not only to calculate gridded concentrations but also concentrations at point re-ceptors, by means of suitable kernels sampling particles in a certain neighbourhood of the receptor. This can be used to obtain s–r relationships for point sources if the (candidate) source locations are known.

To convert gridded concentrations resulting from standard forward simulations into s–r relationships with the dimension of time, a transformation is necessary. For source and recep-tor both in mass units, it reads as follows (for other cases, factors as indicated in Table 1 need to applied):

∂ ¯χ ∂ ˙qin = 1TsVs µt ot ¯ c, (18)

where 1Tsis the time during which the (constant!) source is

acting, Vs is its volume and ¯c is the concentration in a grid

cell as delivered by the forward run. For Flexpart, the above value must be multiplied with the factor 10−12, because re-sults are scaled by that factor.

2.5 Dry and wet deposition

Source-receptor relationships are also of interest with respect to receptors representing accumulated deposition. The de-position fluxes are obtained by linear operators on the the concentration field (in an LPDM, on the masses of parti-cles). Therefore it is easy to extend the methods presented to this case. In an efficient implementation, pseudo-sources with strengths varying as a function of time and space, pro-portional to the deposition velocity or wash-out coefficients, would be used. This has not yet been implemented in Flex-part.

2.6 Convection

Most Lagrangian particle dispersion models do not simulate the effects of moist (i.e., cloud) convection. However, a pa-rameterisation for convective transports has been introduced recently into Flexpart (Seibert et al., 2002). The scheme re-distributes particles from their initial level to a randomly cho-sen destination level. The transition probabilities are given in discrete form in a so-called redistribution matrix which is calculated from the temperature and humidity profiles. Be-cause convection manifests as concentrated updrafts with a high vertical velocity and weak compensating subsidence oc-cupying a larger area, the redistribution matrix is not sym-metric. In a backward run, particles must be redistributed from the destination level back to the initial level. Thus, the transposed redistribution matrix has to be used. This means that the probability of a particle of having arrived at its present level from another level is considered.

Another issue which requires attention is the part of the scheme which calculates the redistribution matrix. A one-dimensional convection model (Emanuel and ˇ Zivkovi´c-Rothman, 1999), designed for use in a prognostic meteoro-logical model, is run for each grid column containing parti-cles. The model is integrated forward in time for a few hours, calculating Eulerian tracer tendencies due to the action of

P. Seibert and A. Frank: Source-receptor matrix calculation with a Lagrangian particle dispersion model 57 convection, from which the redistribution matrix is derived.

Tendencies of the grid-scale temperature and humidity (taken from the meteorological input to Flexpart) enter this simula-tion, and these tendencies must be forward in time, as the convection submodel must always run forward in time, even in an LPDM backward simulation. The convection subrou-tines are called only once for each set of meteorological in-put fields, thus typically every 3 or 6 h, in the middle of the time between the two fields (this has been changed in Flex-part recently). If a sampling or release period is not dis-tributed symmetrically around the time when convection is calculated, differences between forward and backward mode modes will arise. They are an unavoidable consequence of the coarse temporal discretisation for the convection, which, however, is necessary because the convection simulation is so time-consuming.

3 Test calculations

Source-receptor relationships have been calculated both in forward and backward mode for a number of test cases. The purpose of these test calculations was to illustrate the con-cept and to have empirical support for the correctness of the derivations presented in the first part of this paper. During this work it turned out that these tests – comparison between forward and backward simulations as well as calculations for which the result can be analytically derived – is also a very useful validation check for a model. In Flexpart, we found a few bugs and model limitations in this process, mainly re-lated to the numerical implementation of wet scavenging and to the release and sampling of particles. Most of these bugs would not affect standard applications in a significant way.

The tests are described here in the order of increasing com-plexity. They were carried out in final form with Flexpart Version 5; unless otherwise mentioned, bugs and limitations found were corrected for the tests. The next version of Flex-part will include all the improvements worked out during these tests. More details will be provided through the Flex-part home page on the world-wide web (cf. Footnote 1).

Presently, particles are released in standard forward mode of Flexpart equally distributed throughout the vertical exten-sion of the source. It would be more realistic to release them with the same profile as air density, but again, as long as the layer is not too deep, the difference is not a big one.

3.1 Test 1: Release and sampling

In this test, particles representing a conservative tracer are released during one day in a certain volume which is at the same time the sampling volume. It was chosen to be 1◦×1◦×500 m large. All velocities (mean wind and

turbu-Table 2. Results of release and sampling test.

type of calculation s–r relationship [s]

analytical 43,200

forward 43,233

backward 43,233

Number of particles: 1000

Synchronisation time interval: 300 s

lence) are set to zero, and the sampling kernel (attributing the mass of each particle to the four surrounding grid cells) was switched off so that particles are attributed completely to the cell where they are located. Thus, the only processes present in the model were release and sampling of particles, and any errors are due to the discretisation of these processes. (Therefore we can work with a relative small number of parti-cles.) The sampling took place during the same 24 h in which the source was active. Thus, the analytical s–r relationship is the mean residence time of the particles in the volume, 43 200 s. Test results and some additional characteristics are reported in Table 2. There is no difference between forward and backward mode. The small difference (less than 1 per mille) between numerical and analytical result is caused by an inaccurate temporal discretisation of release and sampling of particles, affecting the first and last intervals only. 3.2 Test 2: Wet scavenging

This test is the same as Test 1, except that a strong wet scavenging is switched on with a prescribed, linearly vary-ing precipitation rate. In the available versions of Flexpart, wet scavenging acts on all particles regardless of their height. Thus, the true s–r relationship could be calculated analyti-cally. Scavenging is implemented in Flexpart as a scavenging rate which is a nonlinear function of the precipitation inten-sity (Stohl and Seibert, 2002). Furthermore, tests were con-ducted for stationary precipitation fields with convective cipitation only, large-scale precipitation only, or mixed pre-cipitation, all with a total cloud cover of 1. In all these tests, differences were at most a few seconds (Table 3).

3.3 Test 3: Realistic scenario for short-distance transport In this test, particles were released in the same grid volume as in the previous tests, however, they were sampled in an-other grid volume two degrees to the south-east, during the same 24 h as the release. The co-ordinates of the grid cen-tres were 57◦N, 20◦E and 59◦N, 18◦E. Real meteorological fields (ECMWF analysis, 1◦horizontal resolution, 60 layers, every 3 h) were used as input, and three different types of tracers were simulated: conservative, short-lived (“radioac-tive noble gas”, half-life 0.5 d), and a tracer subject to wet

0 1000 2000 3000 4000 s−r relationship FWD [s] −200 −100 0 100 200 s−r relationsip difference FWD − BWD [s] 5% 1% 10%

300 s, all on (normal Flexpart)

300 s, Flexpart without turbulence and kernel 300 s, 20 boxes, 400,000 particles 300 s, 500 boxes, 1,000,000 particles 900 s, 500 boxes, 1,000,000 particles

Fig. 2. Scatter plot of s–r relationships [s] for the realistic

short-range transport of a conservative tracer (Test 3). The differences be-tween forward and backward calculation results are plotted against the forward results. Each dot represents one day. Dashed lines in-dicate relative deviations between the two calculation modes. The calculations were carried out with different time steps and parti-cle numbers (partly released in sub-boxes of the source for more homogeneous initial distribution) as indicated in the insert. All cal-culations except the “normal Flexpart” are without turbulence and kernel and, unless otherwise indicated, with 40 000 particles.

scavenging (“aerosol”, scavenging properties as in Test 2). This test was carried out for the days 1–20 October 2000. Convection was switched off (note that this refers only to vertical redistribution through moist convection, convective precipitation is taken from ECMWF analysis and is always considered). In this case, fluctuations of the concentrations occur as a result of the stochastic nature of the LPDM.

Figure 1 illustrates the test as well as the forward and back-ward calculations modes for the s–r relationship in general. In the forward mode, particles are released from the area of the light square, which is the source in the s–r relationship considered in this test, and tracked as they are transported by the atmospheric flow. The difference of this mode to regu-lar dispersion modelling is only that results have been scaled according to Eq. (18). In the backward mode, particles are released from the area of the receptor in the s–r relationship (which is marked by the bold square in the forward-mode picture and the light square in the backward-mode picture) and tracked backward in time, to their regions of origin. In each calculation mode, only one sampling volume, indicated by the bold square, may be compared with the other mode.

Table 3. Results of wet scavenging test.

type of calculation s–r relationship [s]

analytical 2668

forward 2666

backward 2664

Number of particles: 1000

Synchronisation time interval: 300 s

Scavenging parameters: a = 2 × 10−4, b =0.8 mm−1h Precipitation rate [mm h−1]: 1.9 + 0.1 INT (time / 300 s)

Figure 2 compares the s–r relationships calculated in for-ward and backfor-ward mode with different set-ups of the model. We notice that stochastic effects of turbulence produce devi-ations between both calculation modes. In one set-up, all turbulent components were set to zero. However, even then there are deviations. It was found that they can be dimin-ished by using a large number of particles, and ensuring a homogeneous initial distribution by dividing the source into separate boxes. However, even then some deviations remain, and they decrease when shorter time steps are used. They are an expression of the inherent uncertainties of the trajectory calculation caused by the need to interpolate wind fields to particle positions.

This was corroborated by further tests. With artificial, con-stant wind fields no deviations occurred. One-dimensional trajectories were calculated with the numerical scheme of Flexpart in an analytical, nonlinear wind field. In the first case, velocities were calculated directly from the analytical formula at each position and in the second case they were interpolated from values at a regular grid. For time steps go-ing towards zero, errors were found to go to zero as expected only in the first case, but not in the second one.

This is a further example that the forward-backward com-parison is a useful model test in general, and it might even be used as to estimate the total interpolation error in the results. No differences in behaviour, as far as equality between for-ward and backfor-ward simulations is concerned, were found be-tween species.

3.4 Test 4: Idealised convective redistribution

An idealised test was devised for the cloud convection scheme, assuming complete mixing through moist convec-tion. The redistribution matrix for the convection was defined analytically so that it would lead to complete vertical mixing within one step. The vertical levels used in this matrix were ECMWF levels from a real case, and 50 000 particles were used. Tests were conducted with instantaneous, local sources and results are scaled to be valid for a one square-metre col-umn. This test serves not only as a test for the backward

P. Seibert and A. Frank: Source-receptor matrix calculation with a Lagrangian particle dispersion model 59

P. Seibert and A. Frank: Source-receptor matrix calculation with a Lagrangian particle dispersion model

9

been included with each equation for convenience. For a

col-umn of 1 m

_,

where

is the atmospheric pressure and

is the gravity

ac-celeration.

The test results are shown in Figure 3. Apart from minor

fluctuations which are caused by the stochastic redistribution

of the particles by the convection scheme, analytical and

nu-merical s–r relationships agree. It is left to the reader to

re-flect about the different shapes of the curves and how they are

related to the forward or backward calculation modes and the

different units for source and receptor.

4 Application examples

The new method has already been applied in a study related

to the global radionuclide measurement network established

as a part of the monitoring system for the Comprehensive

Nuclear Test Ban Treaty (Seibert and Frank, 2001), and for

the interpretation of aircraft measurements of long-range

pol-lutant transport in the free atmosphere (Stohl et al., 2002).

In addition, a small example shall be presented here.

Sta-tions monitoring ambient radioactivity in Europe

occasion-ally measure elevated levels of

_{Cs since the Chernobyl}

dis-aster. It appears likely that this radioactive material comes

from resuspension in heavily contaminated areas. We have

done some calculations to test this hypothesis for the

_Cs

monitoring data of October 2000 in Stockholm, Sweden.

Backward as well as forward s–r relationships have been

cal-culated for the measurements as receptors and the area

con-taminated with more than 100 kBq m

_{according to}

Euro-pean Communities et al. (2001). Wet and dry deposition were

considered in the simulations which are based on ECMWF

analyses and – for precipitation and better temporal

resolu-tion – short-range forecasts, with 1

horizontal and 3 h

tem-poral resolution. The simulations were carried out for

trans-port times up to 7 days. While the receptors have a

resolu-tion of 1 to 3 days (depending on the filter change scheme),

the source has been temporally integrated over the whole 7 d

transport time. As receptor, a 1

1

grid mesh was used,

and the height of both source and receptor grid boxes was

300 m. Simulations were carried out with 500 000 particles.

Figure 4 shows that both forward and backward

simula-tions reproduce the major ups and downs of the

observa-tions. Strong inhomogeneities of the contamination within

the source areas, potential variations in resuspension

coeffi-cients and the measurement error (approximated as the

min-imum detectable activity, varying between 2 and 7

–

Bq m

₎

limit the achievable agreement. On some days, there is a

con-siderable difference between forward and backward

simula-tions which we would attribute to general modelling errors,

Receptor height [km a.g.l.]

s-r relationship [10^-4] fwd_mass-mass 0 5 10 15 20 0 .5 1.0 1.5 analytical 0.5 km numerical 0.5 km analytical 9.5 km numerical 9.5 km 22-Apr-03 20:51:56

Source height [km a.g.l.]

s-r relationship [10^-4] bwd_mass-mass 0 5 10 15 20 0 .5 1.0 1.5 analytical numerical analytical numerical 22-Apr-03 20:35:10

Receptor height [km a.g.l.]

s-r relationship [10^-4] fwd_mass-mix 0 5 10 15 20 0 .5 1.0 1.5 analytical 0.5 km numerical 0.5 km analytical 9.5 km numerical 9.5 km 22-Apr-03 20:51:56

Source height [km a.g.l.]

s-r relationship [10^-4] bwd_mass-mix 0 5 10 15 20 0 .5 1.0 1.5 analytical numerical analytical numerical 22-Apr-03 20:35:10

Receptor height [km a.g.l.]

s-r relationship [10^-4] fwd_mix-mass 0 5 10 15 20 0 .5 1.0 1.5 analytical 0.5 km numerical 0.5 km analytical 9.5 km numerical 9.5 km 22-Apr-03 20:51:56

Source height [km a.g.l.]

s-r relationship [10^-4] bwd_mix-mass 0 5 10 15 20 0 .5 1.0 1.5 analytical numerical analytical numerical 22-Apr-03 20:35:11

Receptor height [km a.g.l.]

s-r relationship [10^-4] fwd_mix-mix 0 5 10 15 20 0 .5 1.0 1.5 analytical 0.5 km numerical 0.5 km analytical 9.5 km numerical 9.5 km 22-Apr-03 20:51:56

Source height [km a.g.l.]

s-r relationship [10^-4] bwd_mix-mix 0 5 10 15 20 0 .5 1.0 1.5 analytical numerical analytical numerical 22-Apr-03 20:35:11

Receptor height [km a.g.l.]

s-r relationship [10^-4] fwd_mass-mass 0 5 10 15 20 0 .5 1.0 1.5 analytical 0.5 km numerical 0.5 km analytical 9.5 km numerical 9.5 km 22-Apr-03 20:51:56 Fig. 3. Analytical (red lines) and numerical (black lines) s–r rela-tionships for idealised convection tests in a 1 m2_{air column. Units}

are according to Table 1, instantaneous local source. The left col-umn of graphs shows forward simulations with s–r relationships as a function of the receptor height for two different source heights (thin solid: 0.5 km, heavy broken: 9.5 km). The right column shows backward simulations with s–r relationships as a function of the source height for two different receptor heights (colours corre-spondingly). Forward and backward calculations should agree for these two specific heights.

mainly due to interpolation. The sample of October 12/13

is an unexplained outlier. Apart from this case, the

contam-inated areas around Chernobyl appear to be sufficient to

ex-www.atmos-chem-phys.org/0000/0001/

Atmos. Chem. Phys., 0000, 0001–12, 2003

Fig. 3. Analytical (red lines) and numerical (black lines) s–r relationships for idealised convection tests in a 1 m2_{air column. Units are}

according to Table 1, instantaneous local source. The left column of graphs shows forward simulations with s–r relationships as a function of the receptor height for two different source heights (thin solid: 0.5 km, heavy broken: 9.5 km). The right column shows backward simulations with s–r relationships as a function of the source height for two different receptor heights (colours correspondingly). Forward and backward calculations should agree for these two specific heights.

8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 day number 0 5 10 15 20 Cs−137 [uBq/m3] fwd model: 0.51 + 1.00 m (r = 0.74) bwd model: 0.05 + 0.99 m (r = 0.94) obs error bars

observation

Observed and modelled Cs−137 concentration

Contaminated areas − Stockholm, October 2000

8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 day number 0 5 10 15 20 Cs−137 [uBq/m3] fwd model: 1.37 + 37 m (r = 0.69) bwd model: 0.50 + 45 m (r = 0.81) obs error bars

observation

Observed and modelled Cs−137 concentration

Chernobyl − Stockholm, October 2000

Fig. 4. Time series of the observed (with error margins) and

mod-elled concentration of Cs-137 at Stockholm for a part of October 2000. The model is a simple regression between observations and s–r relationship, calculated in either forward or backward mode, between Stockholm and the Chernobyl area. The regression for-mulae and resulting correlation coefficients are given in the legend; both are determined without the measurement of October 13th (out-lier). The top panel is for a source covering the contaminated areas in Ukraine, Belorus and Russia as indicated in Fig. 5, the bottom panel for a source area limited to the grid element containing the Chernobly power plant (0.5◦size).

method, but also as a test for the multipliers given in Table 1, because of the strong differences in ambient pressure. We don’t compare forward and backward calculations directly, but rather we compare both of them with the analytical so-lution. This way of presentation illustrates also the different nature of forward (source-oriented) and backward (receptor-oriented) calculations. Each forward calculation gives the s–r

relationship for one source and the whole profile of receptors, whereas each backward calculation is for one receptor and the whole source profile. Two calculations were performed for each mode, with a low-level and a high-level release. If we denote the source height by z and the receptor height by ζ, and the release in mass units by µ and in mixing ratio units by ν, the analytical solutions for the s–r relationships read:

mass – mass: ∂c(z, ζ ) ∂µ = ρ(ζ ) MA (19) mass – mix: ∂χ (z, ζ ) ∂µ = 1 MA (20) mix – mass: ∂c(z, ζ ) ∂ν = ρ(z)ρ(ζ ) MA (21) mix – mix: ∂χ (z, ζ ) ∂ν = ρ(z) MA (22)

where c and χ are the mass concentration and the mass mix-ing ratio as previously, and MA is the total mass of the air

within the column that is being completely mixed. The short source and receptor unit characterisations from Table 1 have been included with each equation for convenience. For a col-umn of 1 m2_,

MA =(pbottom−ptop) g−1,

where p is the atmospheric pressure and g is the gravity ac-celeration.

The test results are shown in Fig. 3. Apart from minor fluc-tuations which are caused by the stochastic redistribution of the particles by the convection scheme, analytical and numer-ical s–r relationships agree. It is left to the reader to reflect about the different shapes of the curves and how they are re-lated to the forward or backward calculation modes and the different units for source and receptor.

4 Application examples

The new method has already been applied in a study related to the global radionuclide measurement network established as a part of the monitoring system for the Comprehensive Nuclear Test Ban Treaty (Seibert and Frank, 2001), and for the interpretation of aircraft measurements of long-range pol-lutant transport in the free atmosphere (Stohl et al., 2002).

In addition, a small example shall be presented here. Sta-tions monitoring ambient radioactivity in Europe occasion-ally measure elevated levels of137_{Cs since the Chernobyl}

dis-aster. It appears likely that this radioactive material comes

P. Seibert and A. Frank: Source-receptor matrix calculation with a Lagrangian particle dispersion model 61 from resuspension in heavily contaminated areas. We have

done some calculations to test this hypothesis for the137_Cs

monitoring data of October 2000 in Stockholm, Sweden. Backward as well as forward s–r relationships have been cal-culated for these measurements as receptors. As the source, we considered either the areas in Ukraine, Belorus and Rus-sia contaminated with more than 100 kBq m-2_{according to}

European Communities et al. (2001), or the single 0.5◦grid cell containing the Chernobyl power plant. Wet and dry de-position were considered in the simulations which are based on ECMWF analyses and – for precipitation and better tem-poral resolution – short-range forecasts, with 1◦ horizontal and 3 h temporal resolution. The simulations were carried out for transport times up to 7 days. While the receptors have a resolution of 1 to 3 days (depending on the filter change scheme), the source has been temporally integrated over the whole 7 d transport time. As receptor, a 1◦_×1◦_{grid mesh}

was used, and the height of both source and receptor grid boxes was 300 m. Simulations were carried out with 500 000 particles.

Figure 4 (top) shows that both forward and backward sim-ulations reproduce the major ups and downs of the observa-tions. Strong inhomogeneities of the contamination within the source areas, potential variations in resuspension coeffi-cients and the measurement error (approximated as the min-imum detectable activity, varying between 2 and 7 µBq m-3₎

limit the achievable agreement. On some days, there is a con-siderable difference between forward and backward simula-tions which we would attribute to general modelling errors, mainly due to interpolation. The sample of October 12/13 is an outlier. It might be caused by caesium picked up in an area northwest of Stockholm (Fig. 5, bottom), at a dis-tance of less than 100 km, contaminated with values in the 40-100 kBq m-2_{range (European Communities et al., 2001).}

Leaving this case out, the model based on the backward sim-ulation results in a correlation coefficient of 0.94 and the expected direct proportionality between s–r relationship and concentration (regression constant 0.05). Interestingly, the model based on the forward simulation performs worse (cor-relation coefficient of 0.74).

If only the area immediately around the reactor is consid-ered as potential source (Fig. 4, bottom), the correlation co-efficient drops for both the forward and the backward simula-tion, indicating that it is indeed more likely that the observed caesium originates from resuspension than from ventings of the sarcophagus in which the reactor has been enclosed. Re-placing the larger source area with the small one so that both source and receptor are small, the correlation coefficient for the backward model drops much stronger than for the for-ward model, and the difference between forfor-ward an back-ward models is reduced (especially if considered in terms of explained variance). This could be a hint that a small release area for the computational particles combined with a large re-ceptor area (backward simulation for contaminated areas) is

P. Seibert and A. Frank: Source-receptor matrix calculation with a Lagrangian particle dispersion model 11

Fig. 5. Some examples of s–r relationships for the receptor at Stock-holm as determined in the backward mode for different measure-ment periods. The s–r relationship is for an area source and has units of s m-1_{. The contaminated areas around Chernobyl}

consid-ered for the evaluation in Figure 4 are indicated by their perimeters and the Chernobyl power plant site by a black dot. The receptor at Stockholm is marked by a square. Political borders are those of 1986. A geographical grid with a mesh size of 10¹ has been

over-laid; the southwestern corner is at 40¹ N, 10¹ E. The figures refer to

the sampling periods as follows: top: ending on 9 Oct, middle: 11 Oct, bottom: 13 Oct (the outlier).

additional motivation. His contributions to discussions helped to clarify a number of issues in the development of the

methodol-ogy. Meteorological fields for test calculations were provided by the ECMWF in the frame of the Special Project ‘SPATMOT’. Ra-dionuclide monitoring data from Stockholm were kindly supplied by the Swedish Defense Research Agency (Ingemar Vintersved).

References

Emanuel, K. A. and ˇZivkovi´c-Rothman (1999), Development and evalution of a convection scheme for use in climate models. J. Atmos. Sci.56, 1766–1782.

European Communities, Roshydromet, Minchernobyl, and Belhy-dromet (2001), Atlas of caesium deposition on Europe after the Chernobyl accident. Luxembourg, Office for Official Publication of the EU, ISBN 92-894-1004-3.

Flesch, T. K., J. D. Wilson, and E. Lee (1995), Backward-time la-grangian stochastic dispersion models and their application to es-timate gaseous emissions. J. Appl. Meteorol.34, 1320–1333. Giering, R. (1999), Tangent linear and adjoint bigeochemical

mod-els. In: P. Kasibhatla, M. Heimann, P. Rayner, N. Mahowald, R. G. Prinn, and D. E. Hartley (eds.), Inverse Methods in Global Biogeochemical Cycles, pp. 33–48, AGU Geophysical Mono-graph Vol. 114, ISBN 0-87590-097-6, Washington.

Hourdin, F. and J.-P. Issartel (2000), Sub-surface nuclear tests mon-itoring through the CTBT xenon network. Geophys. Res. Let. 127(15), 2245–2248.

Robertson, L. and C. Persson (1992), On the application of four-dimensional data assimilation of air pollution data using the ajoint technique. In: H. van Dop and G. Kallos (eds.), Air Pollu-tion Modeling and its ApplicaPollu-tion, vol. IX, pp. 365–373, Penum Press, New York.

Seibert, P. (1999), Inverse modelling of sulfur emissions in Europe based on trajectories. In: P. Kasibhatla, M. Heimann, P. Rayner, N. Mahowald, R. . G. Prinn, and D. E. Hartley (eds.), Inverse Methods in Global Biogeochemical Cycles, pp. 147–154, AGU Geophysical Monograph Vol. 114, ISBN 0-87590-097-6. Seibert, P. (2001), Inverse modelling with a Lagrangian particle

dis-persion model: application to point releases over limited time intervals. In: F. Schiermeier and S.-E. Gryning (eds.), Air Pol-lution Modeling and its Application XIV, pp. 381–390, Kluwer Academic Publ.

Seibert, P. and A. Frank (2001), Backward atmospheric transport and dispersion modelling with FLEXPART in the context of eval-uation of atmospheric transport modelling tools at CTBTO/PTS. In: Evaluation of Atmospheric Transport Modelling Tools Used at the Provisional Technical Secretariat, Annex 6. 8 pp. Seibert, P., B. Kr¨uger, and A. Frank (2002), Parametrisation of

con-vective mixing in a Lagrangian particle dispersion model. In: Proceedings of the 5th GLOREAM Workshop, Wengen, Switzer-land.

Stohl, A., S. Eckhardt, C. Forster, P. James, N. Spichtinger, and P. Seibert (2002), A replacement for simple back trajectory cal-culations in the interpretation of atmospheric trace substance measurements. Atmos. Environ.36(29), 4635–4648.

Stohl, A., M. Hittenberger, and G. Wotawa (1998), Validation of the Lagrangian particle dispersion model Flexpart against

large-scale tracer experiment data. Atmos. Environ. 32(24), 4245–

4264.

Stohl, A. and P. Seibert (2002), The Flexpart particle dis-persion model, Version 5.0, User Guide. On-line at

www.atmos-chem-phys.org/0000/0001/ Atmos. Chem. Phys., 0000, 0001–12, 2003

Fig. 5. Some examples of s–r relationships for the receptor at

Stock-holm as determined in the backward mode for different measure-ment periods. The s–r relationship is for an area source and has units of s m-1_{. The contaminated areas around Chernobyl}

consid-ered for the evaluation in Fig. 4 are indicated by their perimeters and the Chernobyl power plant site by a black dot. The receptor at Stockholm is marked by a square. Political borders are those of 1986. A geographical grid with a mesh size of 10◦has been over-laid; the southwestern corner is at 40◦N, 10◦E. The figures refer to the sampling periods as follows: top: ending on 9 October, middle: 11 October, bottom: 13 October (the outlier).

less susceptible to numerical errors than vice versa (forward simulation for contaminated areas).

5 Outlook

The method introduced in this paper has the potential to su-persede back trajectories for the quantitative interpretation of atmospheric trace substance measurements. For more qual-itative interpretation, LPDM backward calculations with a trajectory-like compressed output as suggested by Stohl et al. (2002) can be recommended. Our vision is that a background monitoring station would be associated with an archive of the s–r data for each measurement. To keep the archive at a reasonable size, the source vector would probably be grid-ded only in space but not in time, with one or a few layers vertically and a grid resolution which is finer near the sta-tion and coarser at longer distances. Together with a small software tool and an emission data base, the modelled con-tributions of selectable source regions to the station could be quickly calculated and compared to measurements. Devia-tions could be analysed in various ways, including inverse modelling (see, e.g. Seibert, 1999). A map of the (2-d or 3-d) s–r relationship (directly or multiplied with emissions) could be termed a “footprint” of that station in analogy to the micro-meteorological flux “footprints”. We admit, how-ever, that this wording is not ideal as, rather, emissions leave a kind of “footprint” in the atmosphere, and not vice versa. We suggest an alternative term: the s–r relationship as the “field of view” of a station.

Of course, there are also other methods to calculate such s–r climatologies. The advantage of the backward method is obvious for a receptor-oriented view. Compared to an adjoint Eulerian model, the advantage of the LPDM is that there is no initial diffusion due to the release of the adjoint tracer into a finite-size grid cell, which gives the LPDM an advantage as a measurement station acts like a point source in a backward simulation. Furthermore, long-range transport can be simu-lated more accurately as no artificial numerical diffusion is present. Especially when filaments created by deformation processes become small against the mesh size, Eulerian mod-els reach their limits. There are two drawbacks of the LPDM: it cannot simulate nonlinear chemistry, and for very high di-lution ratios, the necessary trade-off between particle number and computational requirements will lead to statistically un-certain results (therefore a coarser grid may be required).

6 Conclusions

A method has been derived and introduced with various ex-amples allowing to calculate relationships of substances not undergoing nonlinear chemical changes with a

backward-running Lagrangian particle dispersion model. This includes situations with wet and dry deposition or any other first-order sources and sinks. Receptor-oriented problems, espe-cially related to point or line (aircraft) measurements, can be treated efficiently and relatively accurately by this method.

The method has been tested extensively with the Flexpart (Stohl et al., 1998) model, and useful extensions have been implemented in this model based on experiences gained in these tests. It was also found that many of these tests are generally useful for testing model quality. Differences be-tween forward and backward simulations in realistic condi-tions may be a useful tool to obtain one estimate of model errors, which is quite important for a subsequent inversion to derive sources.

Acknowledgements. This work was supported by the Austrian

Sci-ence Fund (FWF) through project P12595 “Inverse modelling of atmospheric trace constituents with Lagrangian models on the Eu-ropean scale”, and it is a contribution to the EUROTRAC-2 subpro-jects GLOREAM and GENEMIS-2. A part of the manuscript was written during a stay of the first author at the Canadian Meteorolog-ical Centre (CMC), Montr´eal, Canada; the support of the Meteoro-logical Service of Canada and the warm reception by the colleagues at CMC is gratefully acknowledged. The continuous co-operation with A. Stohl (Technical University of Munich, Germany), main au-thor of Flexpart, was important for the integration of the backward calculation mode into this model and provided additional motiva-tion. His contributions to discussions helped to clarify a number of issues in the development of the methodology. Meteorological fields for test calculations were provided by the ECMWF in the frame of the Special Project “SPATMOT”. Radionuclide monitor-ing data from Stockholm were kindly supplied by the Swedish De-fense Research Agency (I. Vintersved).

References

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